Will I be sinful if I remain estranged from my friend, even though I bear her no grudge?

Islam encourages love and communication and prohibits animosity. The Prophet, peace and blessings be upon him, forbade a Muslim from shunning his brother for more than three nights and condemned two brothers who cut ties. It is not permissible to shun a Muslim except for a legitimate Shar'i reason. Therefore, there is no justification for boycotting your friend. Your apology to her and her forgiveness of you are sufficient. You should be the better person and initiate the greeting, reminding her of the seriousness of shunning. You will not be sinful if she insists on the boycott.